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314       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

(a)   Let A be a Banach algebra, and put b = inf (\\x2 \\l\\x\\2). Show that

**0    X

for all x E A.

(b)   If x, y are two elements of a Banach algebra A, show that the intersections of

Sp(xy) and $p(yx) with C  {0} are equal, and hence that p(xy) = p(yx) (cf. Section
11.1, Problem 2).

2.    Let A be a Banach algebra with unit element.

(a)    If x e A has a left (resp. right) inverse y, then each x' e A such that

I* - x\\ <\\y\\-1

has a left (resp. right) inverse.

(b)    Let (xn) be a sequence of elements of A, each of which has a left (resp. right)
inverse yn . If the sequence (#) converges to x, and if the sequence (yn) is bounded,
then x has a left (resp. right) inverse.

3.    Let A be a normed algebra. For each x e A, let Lx (resp. Rx) denote the continuous
linear mapping y \-+xy (resp. y\-*yx) of A into A. The element x is said to be a left
(resp. right) topological zero-divisor if Lx (resp. Rx) is not a homeomorphism of A
onto its image.

(a)    Give an example of a left topological zero-divisor which is not a left zero-divisor
(cf. Section 11.1, Problem 4).

(b)    For each x e A, put

A(x) = inf ||xy||/|M|,       A'(x) = inf \\yx\\l\\y\\.

y#Q                                                       y?fcO

Show that

|A(*) - AW! g || x~ yl                |A'(jc) - A'GOI < |jjc - y\\,

X(x)\(y)

Deduce that the set of left (resp. right) topological zero-divisors is closed in A.

(c)    Let A be a Banach algebra with unit element. If x e A is not left-invertible but
is the limit of a sequence (xn) of left-invertible elements, then x is a right topological
zero divisor (use Problem 2(b)). Deduce that the set of elements of A which are neither
invertible nor topological zero-divisors is open in A.

(d)    In the algebra jtf(X) (15.15), show that the identity function lx is neither
invertible nor a topologicai zero-divisor.

4.    Let A be a Banach algebra with unit element e, in which the only right topological
zero-divisor is 0. Show that A == Ce. (Consider the frontier points of Sp(x) and use
Problem 3(c).)

5.    (a)   In a Banach algebra A with unit element, an element x is said to be topologically
nilpotent if the sequence (xn)n^1 tends to 0. Show that x is topologically nilpotent
if and only if p(x) < 1.

(b)    An element x e A is said to be quasi-nilpotent if p(x) = 0, or equivalently if
(A*)" tends to 0 as n tends to 4- <x>, for all scalars A. A quasi-nilpotent element is a left
and right topological zero-divisor.

(c)    Let u be a continuous endomorphism of a Banach space E. If lim ||w"(f)|| 1/fl = 0

rt-00

for all / e E, show that u is quasi-nilpotent in the Banach algebra A = -S?(E). (Observeugh positive values, >(FM(x0 4- /) tends to a limit equal to 7rg(x0).d. Then, for each neighborhood U of e, there exists a number e > 0 such that